Quasiregular figures Right triangle domains (p q 2), = r{p,q} r{4,3} r{5,3} r{6,3} r{7,3}... r{∞,3} (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.∞)2 Isosceles triangle domains (p p 3), = = h{6,p} h{6,4} h{6,5} h{6,6} h{6,7}... h{6,∞} = = = = = (4.3)4 (5.3)5 (6.3)6 (7.3)7 (∞.3)∞ Isosceles triangle domains (p p 4), = = h{8,p} h{8,3} h{8,5} h{8,6} h{8,7}... h{8,∞} = = = = = (4.3)3 (4.5)5 (4.6)6 (4.7)7 (4.∞)∞ Scalene triangle domain (5 4 3), (3.5)4 (4.5)3 (3.4)5 Aquasiregular polyhedron ortiling has exactly two kinds of regular face, which alternate around each vertex. Theirvertex figures areisogonal polygons.

Regular and quasiregular figures Right triangle domains (p p 2), = = r{p,p} = {p,4}1⁄2 {3,4}1⁄2 r{3,3} {4,4}1⁄2 r{4,4} {5,4}1⁄2 r{5,5} {6,4}1⁄2 r{6,6}... {∞,4}1⁄2 r{∞,∞} = = = = = (3.3)2 (4.4)2 (5.5)2 (6.6)2 (∞.∞)2 Isosceles triangle domains (p p 3), = = {p,6}1⁄2 {3,6}1⁄2 {4,6}1⁄2 {5,6}1⁄2 {6,6}1⁄2... {∞,6}1⁄2 = = = = = (3.3)3 (4.4)3 (5.5)3 (6.6)3 (∞.∞)3 Isosceles triangle domains (p p 4), = = {p,8}1⁄2 {3,8}1⁄2 {4,8}1⁄2 {5,8}1⁄2 {6,8}1⁄2... {∞,8}1⁄2 = = = = = (3.3)4 (4.4)4 (5.5)4 (6.6)4 (∞.∞)4 Aregular polyhedron ortiling can be considered quasiregular if it has an even number of faces around each vertex (and thus can have alternately colored faces).

Ingeometry, aquasiregular polyhedron is auniform polyhedron that has exactly two kinds ofregular faces, which alternate around eachvertex. They arevertex-transitive andedge-transitive, hence a step closer toregular polyhedra than thesemiregular, which are merely vertex-transitive.

Theirdual figures areface-transitive and edge-transitive; they have exactly two kinds of regularvertex figures, which alternate around eachface. They are sometimes also considered quasiregular.

There are only twoconvex quasiregular polyhedra: thecuboctahedron and theicosidodecahedron. Their names, given byKepler, come from recognizing that their faces are all the faces (turned differently) of thedual-paircube andoctahedron, in the first case, and of the dual-pairicosahedron anddodecahedron, in the second case.

These forms representing a pair of a regular figure and its dual can be given a verticalSchläfli symbol${\displaystyle \left\{\begin{array}{c}p\\ q\end{array}\right\}}$ orr{p,q}, to represent that their faces are all the faces (turned differently) of both the regular{p,q} and the dual regular{q,p}. A quasiregular polyhedron with this symbol will have avertex configurationp.q.p.q (or(p.q)²).

More generally, a quasiregular figure can have avertex configuration(p.q)^r, representingr (2 or more) sequences of the faces around the vertex.

Tilings of the plane can also be quasiregular, specifically thetrihexagonal tiling, with vertex configuration (3.6)².Other quasiregular tilings exist on the hyperbolic plane, like thetriheptagonal tiling, (3.7)². Or more generally:(p.q)², with1/p + 1/q < 1/2.

Regular polyhedra and tilings with an even number of faces at each vertex can also be considered quasiregular by differentiating between faces of the same order, by representing them differently, like coloring them alternately (without defining any surface orientation). A regular figure withSchläfli symbol{p,q} can be considered quasiregular, with vertex configuration(p.p)^q/2, ifq is even.

Examples:

The regularoctahedron, with Schläfli symbol {3,4} and 4 being even, can be considered quasiregular as atetratetrahedron (2 sets of 4 triangles of thetetrahedron), with vertex configuration (3.3)^4/2 = (3_a.3_b)², alternating two colors of triangular faces.

Thesquare tiling, with vertex configuration 4⁴ and 4 being even, can be considered quasiregular, with vertex configuration (4.4)^4/2 = (4_a.4_b)², colored as acheckerboard.

Thetriangular tiling, with vertex configuration 3⁶ and 6 being even, can be considered quasiregular, with vertex configuration (3.3)^6/2 = (3_a.3_b)³, alternating two colors of triangular faces.

Regular (p | 2 q) and quasiregular polyhedra (2 | p q) are created from aWythoff construction with the generator point at one of 3 corners of the fundamental domain. This defines a single edge within the fundamental domain.

Coxeter defines aquasiregular polyhedron as one having aWythoff symbol in the formp | q r, and it is regular if q=2 or q=r.^[1]

TheCoxeter-Dynkin diagram is another symbolic representation that shows the quasiregular relation between the two dual-regular forms:

Schläfli symbol		Coxeter diagram	Wythoff symbol
${\displaystyle \left\{\begin{array}{c}p,q\end{array}\right\}}$	{p,q}		q | 2 p
${\displaystyle \left\{\begin{array}{c}q,p\end{array}\right\}}$	{q,p}		p | 2 q
${\displaystyle \left\{\begin{array}{c}p\\ q\end{array}\right\}}$	r{p,q}	or	2 | p q

There are two uniformconvex quasiregular polyhedra:

1. Thecuboctahedron${\displaystyle \left\{\begin{array}{c}3\\ 4\end{array}\right\}}$, vertex configuration(3.4)²,Coxeter-Dynkin diagram
2. Theicosidodecahedron${\displaystyle \left\{\begin{array}{c}3\\ 5\end{array}\right\}}$, vertex configuration(3.5)²,Coxeter-Dynkin diagram

In addition, theoctahedron, which is alsoregular,${\displaystyle \left\{\begin{array}{c}3\\ 3\end{array}\right\}}$, vertex configuration(3.3)², can be considered quasiregular if alternate faces are given different colors. In this form it is sometimes known as thetetratetrahedron. The remaining convex regular polyhedra have an odd number of faces at each vertex so cannot be colored in a way that preserves edge transitivity. It hasCoxeter-Dynkin diagram

Each of these forms the common core of adual pair ofregular polyhedra. The names of two of these give clues to the associated dual pair: respectivelycube${\displaystyle \cap }$octahedron, andicosahedron${\displaystyle \cap }$dodecahedron. Theoctahedron is the common core of a dual pair oftetrahedra (a compound known as thestella octangula); when derived in this way, theoctahedron is sometimes called thetetratetrahedron, astetrahedron${\displaystyle \cap }$tetrahedron.

Regular	Dual regular	Quasiregular common core	Vertex figure
Tetrahedron {3,3} 3 | 2 3	Tetrahedron {3,3} 3 | 2 3	Tetratetrahedron r{3,3} 2 | 3 3	3.3.3.3
Cube {4,3} 3 | 2 4	Octahedron {3,4} 4 | 2 3	Cuboctahedron r{3,4} 2 | 3 4	3.4.3.4
Dodecahedron {5,3} 3 | 2 5	Icosahedron {3,5} 5 | 2 3	Icosidodecahedron r{3,5} 2 | 3 5	3.5.3.5

Each of these quasiregular polyhedra can be constructed by arectification operation on either regular parent,truncating the vertices fully, until each original edge is reduced to its midpoint.

This sequence continues as thetrihexagonal tiling,vertex figure(3.6)² - aquasiregular tiling based on thetriangular tiling andhexagonal tiling.

Regular	Dual regular	Quasiregular combination	Vertex figure
Hexagonal tiling {6,3} 6 | 2 3	Triangular tiling {3,6} 3 | 2 6	Trihexagonal tiling r{6,3} 2 | 3 6	(3.6)2

Thecheckerboard pattern is a quasiregular coloring of thesquare tiling,vertex figure(4.4)²:

Regular	Dual regular	Quasiregular combination	Vertex figure
{4,4} 4 | 2 4	{4,4} 4 | 2 4	r{4,4} 2 | 4 4	(4.4)2

Thetriangular tiling can also be considered quasiregular, with three sets of alternating triangles at each vertex, (3.3)³:

h{6,3} 3 | 3 3 =

In the hyperbolic plane, this sequence continues further, for example thetriheptagonal tiling,vertex figure(3.7)² - aquasiregular tiling based on theorder-7 triangular tiling andheptagonal tiling.

Regular	Dual regular	Quasiregular combination	Vertex figure
Heptagonal tiling {7,3} 7 | 2 3	Triangular tiling {3,7} 3 | 2 7	Triheptagonal tiling r{3,7} 2 | 3 7	(3.7)2

Coxeter, H.S.M. et al. (1954) also classify certainstar polyhedra, having the same characteristics, as being quasiregular.

Two are based on dual pairs of regularKepler–Poinsot solids, in the same way as for the convex examples:

thegreat icosidodecahedron${\displaystyle \left\{\begin{array}{c}3\\ 5/2\end{array}\right\}}$, and thedodecadodecahedron${\displaystyle \left\{\begin{array}{c}5\\ 5/2\end{array}\right\}}$:

Regular	Dual regular	Quasiregular common core	Vertex figure
Great stellated dodecahedron {5/2,3} 3 | 2 5/2	Great icosahedron {3,5/2} 5/2 | 2 3	Great icosidodecahedron r{3,5/2} 2 | 3 5/2	3.5/2.3.5/2
Small stellated dodecahedron {5/2,5} 5 | 2 5/2	Great dodecahedron {5,5/2} 5/2 | 2 5	Dodecadodecahedron r{5,5/2} 2 | 5 5/2	5.5/2.5.5/2

Nine more are thehemipolyhedra, which arefaceted forms of the aforementioned quasiregular polyhedra derived from rectification of regular polyhedra. These include equatorial faces passing through the centre of the polyhedra:

Quasiregular (rectified)	Tetratetrahedron	Cuboctahedron	Icosidodecahedron	Great icosidodecahedron	Dodecadodecahedron
Quasiregular (hemipolyhedra)	Tetrahemihexahedron 3/2 3 | 2	Octahemioctahedron 3/2 3 | 3	Small icosihemidodecahedron 3/2 3 | 5	Great icosihemidodecahedron 3/2 3 |5/3	Small dodecahemicosahedron 5/35/2 | 3
Vertex figure	3.4.3/2.4	3.6.3/2.6	3.10.3/2.10	3.10/3.3/2.10/3	5/2.6.5/3.6
Quasiregular (hemipolyhedra)		Cubohemioctahedron 4/3 4 | 3	Small dodecahemidodecahedron 5/4 5 | 5	Great dodecahemidodecahedron 5/35/2 |5/3	Great dodecahemicosahedron 5/4 5 | 3
Vertex figure		4.6.4/3.6	5.10.5/4.10	5/2.10/3.5/3.10/3	5.6.5/4.6

Lastly there are threeditrigonal forms, all facetings of the regular dodecahedron, whose vertex figures contain three alternations of the two face types:

Image	Faceted form Wythoff symbol Coxeter diagram	Vertex figure
	Ditrigonal dodecadodecahedron 3 | 5/3 5 or	(5.5/3)3
	Small ditrigonal icosidodecahedron 3 | 5/2 3 or	(3.5/2)3
	Great ditrigonal icosidodecahedron 3/2 | 3 5 or	((3.5)3)/2

In the Euclidean plane, the sequence of hemipolyhedra continues with the following four star tilings, whereapeirogons appear as the aforementioned equatorial polygons:

Original rectified tiling	Edge diagram	Solid	Vertex Config	Wythoff	Symmetry group
Square tiling			4.∞.4/3.∞ 4.∞.-4.∞	4/3 4 | ∞	p4m
Triangular tiling			(3.∞.3.∞.3.∞)/2	3/2 | 3 ∞	p6m
Trihexagonal tiling			6.∞.6/5.∞ 6.∞.-6.∞	6/5 6 | ∞
			∞.3.∞.3/2 ∞.3.∞.-3	3/2 3 | ∞

Some authorities argue that, since the duals of the quasiregular solids share the same symmetries, these duals should be called quasiregular too. But not everybody uses this terminology. These duals are transitive on their edges and faces (but not on their vertices); they are the edge-transitiveCatalan solids. The convex ones are, in corresponding order as above:

1. Therhombic dodecahedron, with twotypes of alternating vertices, 8 with three rhombic faces, and 6 with four rhombic faces.
2. Therhombic triacontahedron, with twotypes of alternating vertices, 20 with three rhombic faces, and 12 with five rhombic faces.

In addition, by duality with the octahedron, thecube, which is usuallyregular, can be made quasiregular if alternate vertices are given different colors.

Theirface configurations are of the form V3.n.3.n, andCoxeter-Dynkin diagram

Cube V(3.3)2	Rhombic dodecahedron V(3.4)2	Rhombic triacontahedron V(3.5)2	Rhombille tiling V(3.6)2	V(3.7)2	V(3.8)2

These three quasiregular duals are also characterised by havingrhombic faces.

This rhombic-faced pattern continues as V(3.6)², therhombille tiling.

In higher dimensions, Coxeter defined a quasiregular polytope or honeycomb to have regular facets and quasiregular vertex figures. It follows that all vertex figures are congruent and that there are two kinds of facets, which alternate.^[2]

In Euclidean 4-space, the regular16-cell can also be seen as quasiregular as an alternatedtesseract, h{4,3,3},Coxeter diagrams: =, composed of alternatingtetrahedron andtetrahedroncells. Itsvertex figure is the quasiregulartetratetrahedron (an octahedron with tetrahedral symmetry),.

The only quasiregular honeycomb in Euclidean 3-space is thealternated cubic honeycomb, h{4,3,4}, Coxeter diagrams: =, composed of alternating tetrahedral andoctahedralcells. Its vertex figure is the quasiregularcuboctahedron,.^[2]

In hyperbolic 3-space, one quasiregular honeycomb is thealternated order-5 cubic honeycomb, h{4,3,5}, Coxeter diagrams: =, composed of alternating tetrahedral andicosahedralcells. Its vertex figure is the quasiregularicosidodecahedron,.^[3] A related paracompactalternated order-6 cubic honeycomb, h{4,3,6} has alternating tetrahedral and hexagonal tiling cells with vertex figure is a quasiregulartrihexagonal tiling,.

Quasiregular polychora and honeycombs: h{4,p,q}
Space	Finite	Affine	Compact	Paracompact
Schläfli symbol	h{4,3,3}	h{4,3,4}	h{4,3,5}	h{4,3,6}	h{4,4,3}	h{4,4,4}
	${\displaystyle \left\{3,\genfrac{}{}{0ex}{}{3}{3}\right\}}$	${\displaystyle \left\{3,\genfrac{}{}{0ex}{}{4}{3}\right\}}$	${\displaystyle \left\{3,\genfrac{}{}{0ex}{}{5}{3}\right\}}$	${\displaystyle \left\{3,\genfrac{}{}{0ex}{}{6}{3}\right\}}$	${\displaystyle \left\{4,\genfrac{}{}{0ex}{}{4}{3}\right\}}$	${\displaystyle \left\{4,\genfrac{}{}{0ex}{}{4}{4}\right\}}$
Coxeter diagram	↔	↔	↔	↔	↔	↔
	↔					↔
Image
Vertex figure r{p,3}

Regular polychora or honeycombs of the form {p,3,4} or can have their symmetry cut in half as into quasiregular form, creating alternately colored {p,3} cells. These cases include the Euclideancubic honeycomb {4,3,4} withcubic cells, and compact hyperbolic {5,3,4} withdodecahedral cells, and paracompact {6,3,4} with infinitehexagonal tiling cells. They have four cells around each edge, alternating in 2 colors. Theirvertex figures are quasiregular tetratetrahedra, =.

Regular and Quasiregular honeycombs: {p,3,4} and {p,31,1}
Space	Euclidean 4-space	Euclidean 3-space	Hyperbolic 3-space
Name	{3,3,4} {3,31,1} =${\displaystyle \left\{3,\genfrac{}{}{0ex}{}{3}{3}\right\}}$	{4,3,4} {4,31,1} =${\displaystyle \left\{4,\genfrac{}{}{0ex}{}{3}{3}\right\}}$	{5,3,4} {5,31,1} =${\displaystyle \left\{5,\genfrac{}{}{0ex}{}{3}{3}\right\}}$	{6,3,4} {6,31,1} =${\displaystyle \left\{6,\genfrac{}{}{0ex}{}{3}{3}\right\}}$
Coxeter diagram	=	=	=	=
Image
Cells {p,3}

Similarly regular hyperbolic honeycombs of the form {p,3,6} or can have their symmetry cut in half as into quasiregular form, creating alternately colored {p,3} cells. They have six cells around each edge, alternating in 2 colors. Theirvertex figures are quasiregulartriangular tilings,.

Hyperbolic uniform honeycombs: {p,3,6} and {p,3^[3]}

• v
• t
• e

Form	Paracompact				Noncompact
Name	{3,3,6} {3,3[3]}	{4,3,6} {4,3[3]}	{5,3,6} {5,3[3]}	{6,3,6} {6,3[3]}	{7,3,6} {7,3[3]}	{8,3,6} {8,3[3]}	...{∞,3,6} {∞,3[3]}
Image
Cells	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

• Chiral polytope
• Rectification (geometry)

• Cromwell, P.Polyhedra, Cambridge University Press (1977).
• Coxeter,Regular Polytopes, (3rd edition, 1973), Dover edition,ISBN 0-486-61480-8, 2.3Quasi-Regular Polyhedra. (p. 17), Quasi-regular honeycombs p.69

• Weisstein, Eric W."Quasiregular polyhedron".MathWorld.
• Weisstein, Eric W."Uniform polyhedron".MathWorld. Quasi-regular polyhedra: (p.q)^r
• George Hart,Quasiregular polyhedra

• Quasiregular polyhedra

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